Advanced Analysis

Teacher

KALIPRASAD RATH

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Core Courses

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Course Name : Advanced Analysis

Code(Credit) : CUTM1519(3-0-3)

Course Objectives

  • To point out that iterative processes and convergence of sequences occur in many areas of mathematics, and to develop a general context in metric spaces.
  • To provide a basic course in analysis.
  • To reinforce ideas of proof

Learning Outcomes

  • Understand the Euclidean distance function on Rn and appreciate its properties, and state and use the Triangle and Reverse Triangle Inequalities for the Euclidean distance function on Rn
  • Explain the definition of continuity for functions from Rn to Rm and determine whether a given function from Rn to Rm is continuous
  • Explain the geometric meaning of each of the metric space properties  and be able to verify whether a given distance function is a metric
  • Distinguish between open and closed balls in a metric space and be able to determine them for given metric spaces
  • Define convergence for sequences in a metric space and determine whether a given sequence in a metric space converges
  • State the definition of continuity of a function between two metric spaces.

Course Syllabus

Module-I (T-5hr P-0hr P-6hr)

Metric Spaces: Definition and examples, open and closed spheres,

Project-1: Write a report on applications of metric spaces.

Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.

Project-3: Show that the class of all sequences is a metric space.

Module-II (T-3hr P-0hr P-4hr)

Neighbourhoods, Interior points. Open set

Project-4: Write a report on relation between boundary points, limit points. And their applications.

Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points

Module-III (T-5hr P-0hr P-6hr)

Closed set, boundary points, limit points and isolated points, closure of a set, dense sets.

Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.

Project-7: Write the difference between boundary points, limit points and isolated points explain through examples.

Project-8 (Flip class): Prove theorems on closed set and dense sets.

Module-IV (T-4hr P-0hr P-6hr)

Separable metric spaces, sequences in metric space, convergent sequences, Cauchy sequences

Project-9: Write a report on applications of convergent sequences and Cauchy sequences.

Project-10: Every convergent sequence is bounded. Is the reverse true? Justify your answer with suitable example.

Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.

Module-V (T-4hr P-0hr P-4hr)

Complete metric space, subspace, Cantor’s theorem. Continuous functions,

Project- 12(Flip class): Prove theorems on Complete metric space, subspace.

Project-13(Flip class): : Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.

Module-VI (T-4hr P-0hr P-6hr)

Uniform continuity, Connectedness, connected subsets of separated sets.

Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.

Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example..

Project-16 (Flip class): Prove theorems on connected subsets of separated sets.

Module-VII (T-5hr P-0hr P-4hr)

Disconnected sets, contraction mappings, Banach fixed point theorem.

Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.

Project-18: Write some applications of Banach fixed point theorem.

Text Books:

  1. K. Jain and K. Ahmad, Metric Spaces, Narosa Publishing House, New Delhi.
  2. Methods of Real analysis-R.G. Goldberg.   Chapters: 4(4.2,and 4.3), 5,and  6 (up to 6.8) 

Reference Books:

  1. S.C. Malik and S. Arora – Mathematical Analysis (New Age International)
  2. Principles of Mathematical Analysis – Walter Rudin, III Edition, McGRAW-Hill Book Co.

Session 2

Examples on Metric Spaces

Study Material: http://more examples

Youtube link: https://youtu.be/etP21xln8iQ

Session 3

Project-1: Write a report on applications of metric spaces.

Session 4

Session 5

Project-2: Show that the class denote a set of all bounded sequence of real numbers is a metric space.

Session 6

Open spheres.

youtube link: https://youtu.be/QC2n_Y2XJDM

Session 7

Closed spheres

youtube link: https://youtu.be/QC2n_Y2XJDM

Session 8

Project-3: Show that the class of all sequences is a metric space.

Session 9

Neighbourhood of a point

youtube link: https://youtu.be/pvGWK65vw7A

Session 10

Interior points and theorems

youtube link:https://youtu.be/VsL8Tq8nDow

Session 11

Project-4: Write a report on relation between boundary points, limit points. And their applications

Session 12

Open set

Study material: open set

youtube link: https://youtu.be/EE5FTMBkQjE

Session 13

Project-5(Flip class): Prove theorems on Neighbourhoods, Interior points

Session 14

closed set

study material: closed set and theorems

youtube link: https://youtu.be/PcnSgIoASSk

Session 15

limit points and isolated point

youtube link:https://youtu.be/Z0e4gZZCxBM

Session 16

Project-6: Write a report on difference between open set closed set and limit point , interior point with appropriate examples.

Session 17

closure of a set

youtube link: https://youtu.be/kx7-juddElg

Session 18

Theorems on closurere of a set

youtube link: https://youtu.be/kx7-juddElg

Session 19

dense set

youtube link: https://youtu.be/tI7tC7lkkic

Session 20

Project-7: Write the difference between boundary points, limit points and isolated points explain through examples

Session 21

Project-8 (Flip class): Prove theorems on closed set and dense sets.

Session 22

Separable metric spaces

youtube link: https://youtu.be/lv-j2KK-prc

Session 25

Project-9: Write a report on applications of convergent sequences and Cauchy sequences

Session 26

Project-10: Every convergent sequence is bounded. Is the reverse true? Justify your answer with suitable example.

Session 27

cauchy sequences

youtube link: https://youtu.be/jD5s2jvnH1k

Session 28

Project-11(Flip class): Prove theorems on convergent sequences and Cauchy sequences.

Session 29

subspaces and theorems

youtube link: https://youtu.be/eY-WLtxKpBw

Session 30

Complete metric space

yo utube link:https://youtu.be/X_weB_pYMV4

Session 31

Project-12(Flip class): Prove theorems on Complete metric space, subspace.

Session 32

Session 33

Project- 13: Prove that addition, subtraction, multiplication and division (with condition) of continuous functions are continuous.

Session 34

Theorems on Continuous functions

PPT link: https://www.slideshare.net/VishveshJasani/continuity-of-a-function

Session 35

Uniform continuity and Theorems

youtube lonk: https://youtu.be/IAN7TUbNOCs

Session 36

Connectedness

youtube link: https://youtu.be/_06QIROBDko

Session 37

Project-14(Flip class): Prove theorems on Uniform continuity and Connectedness.

Session 38

Theorems on Connectedness

youtube link: https://youtu.be/mdCU7TlZupI

Session 39

Project- 15: Write the difference between continuity and uniform continuity and explain through a proper example

Session 40

Theorems on Connected and separated sets

youtube link: https://youtu.be/77KtqjqknYk

Session 41

Project-16 (Flip class): Prove theorems on connected subsets of separated sets.

Session 42

Disconnected sets

youtube link: https://youtu.be/mdCU7TlZupI

Session 43

Theorems on Disconnected sets

youtube link : https://youtu.be/VDqhDsT40wU

Session 44

Project- 17(Flip class): Geometrical interpretation of connected and disconnected sets.

Session 45

Banach fixed point theorem.

you tube link: https://youtu.be/9jL8iHw0ans

Session 46

contraction mapping

youtube link: https://youtu.be/vcQSOm2-hEI

Session 47

contraction mapping and theorems

youtube link: https://youtu.be/cwmnjP1rxHM

Session 48

Project-18: Write some applications of Banach fixed point theorem.

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    Kali Prasad Rath received his M.Tech. in Computer Science and  & M.Sc. in Mathematics from Berhampur University.This is his 10th year at Centurion University of Technology and Management and his 17th year in teaching profession. He is associated with qualitative study on theory of difference equation.